Descriptions of neural network architectures that can be used to specify the
generator \(G\)  and the discriminator \(D\)  can be found in the book
of [1]} and in the many references therein.
Often the generator and
discriminator networks would satisfy additional differentiability and moment condition
assumptions. Although such extra assumptions are not necessarily required for
the results that follow, such a `smooth GAN' setting serves as a convenient
example that will be used to illustrate our results. In the following
assumption and in what follows, we use either notation \(\theta \) 
or notation \((\gamma ,\delta )\)  for the elements of \(\Theta =\Gamma \times \Delta \) ;
for instance, \(F(x,z,\theta )\)  and \(F(x,z,\gamma ,\delta )\) 
are used interchangeably.
Also, \(\left|\cdot \right|\)  denotes the Euclidean norm.
